====== Linear case ======

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We found when we [[.:start|relaxed our assumptions]] that the fast dynamics (governed by fast transition reactions between states \(A\) and \(B\)) obeyed either a linear or quadratic rate equation for the frequency of \(A\) types.  Here we investigate the linear case further.  

====== Assumption ======

In this case we assume
<WRAP center round important 60%>
\[
  a_f=0 \text{ or } \nu_{aa} + \nu_{ba}=\nu_{ab} + \nu_{bb}.
\]
</WRAP>

Then
\[
  \diff{f}{t} = b_f f + c_f
\]
and we found the frequency quickly converges to equilibrium at
\[
  f^* = \frac{c_f}{c_f + d_f} = \frac{n \nu_{bb} + \phi_b}{n (\nu_{aa} + \nu_{bb}) + \phi_a + \phi_b}
\]
which always occurs in the range \(0\leq f^* \leq 1\).

====== General Allee ======

On the slow timescale the population follows
\[
\begin{array}{rl}
  n' = & n \left[ f (\tilde{A}_a + \tilde{\alpha}_{ab}) + (1-f) (\tilde{A}_b + \tilde{\alpha}_{ba}) \right] \\
       & - n^2 \left[ f^2 (\tilde{X}_{aa} - \tilde{\beta}_{aab}) + f (1-f) (\tilde{X}_{ab}  + \tilde{X}_{ba}) + (1-f)^2 (\tilde{X}_{bb} - \tilde{\beta}_{bba}) \right]
\end{array}
\]
so, after substitution of \(f=f^*\) we get the general form
\[
  n' = r \, n \, a(n) \left( 1 - \frac{n}{K} \right)
\]
where
\[
\begin{array}{rl}
  a(n) & = 1-A + A \frac{n}{n+n_{1/2}} = \frac{n-n_A}{n+n_{1/2}} \\
  n_A  & \equiv (A-1) n_{1/2}
\end{array}
\]
and the parameters \(r, K, A, \;\&\; n_{1/2}\) can be determined from the rate constants.

We find that resolving two types with a separation of timescales and the assumption \(a_f=0\) the slow dynamics are similar to the Verhulst/logistic equation modulated by a general "Allee" effect. The Allee effect can have different effects, depending on it's "strength", \(A\):
  * Strong Allee, \(A>1\) - below some critical density the population becomes unstable and collapses.
  * Weak Allee, \(0<A \leq 1\) - the carrying-capacity, \(K\), is stable for all densities but the population growth rate is sub-exponential at low densities.
  * No Allee, \(A=0\) - logistic behaviour
  * Contra-Allee, \(A<0\) - low densities favour growth; the population grows super-exponentially at low densities.

====== Examples ======

We are particularly interested in cases with an Allee effect, \(A\neq 0\).  We have already seen both [[.:start#Special case: Only spontaneous transitions|spontaneous]] and [[.:start#Special case: Only induced transitions|induced]] transitions are required so we will focus on these cases below.  In particular, we will assume
<WRAP center round important 60%>
\[
\begin{array}{rcl}
  \nu_{aa}+\nu_{bb} & > & 0 \\
  \phi_a + \phi_b   & > & 0.
\end{array}
\]
</WRAP>

===== Estrous cycle =====

===== Monogamy =====

===== Frequency-independent transitions =====

==== Aggression ====

eg. lemmings' density-dependence

